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Full text of "Analytical Mechanics"

276
ANALYTICAL MECHANICS
one of the vertices perpendicular to the plane of the plate. Find.the resulting angular velocity due to a sudden removal of the axis and a simultaneous introduction of a parallel axis through the center of mass.
8. In the preceding problem, suppose the new axis to pass through one of the other two vertices.
214. Reaction of the Axis of Rotation.—Suppose .B, Fig. 126, to be a rigid body free to rotate about a fixed axis through the point 0, perpendicular to the plane of the figure. If an external force F is applied to the body a part of its action is, in general, transmitted to the axis of rotation. This results in the reaction, R, of the axis, which we will investigate. For the sake of simplicity suppose F to lie in the plane which passes through the center of mass, c, perpendicular to the axis.
Since F and R are supposed to be the only external forces acting upon the body, then by equation (VIII) of p. 242
my = F + R,                                   (1)
where tf is the acceleration of the center of mass. If Fn and FT denote the components of F along and at right angles to the line Oc, respectively, and P and Q the components of R along the same directions, equation (1) may be resolved into the following component-equations:
™fn = Fn+P,                                (2)
mfr = FT+Q,                                (3)
where fn and fr are the components of v. But since the path of the center of mass is a circle
FIG. 126.
7        U
/„ = - =